Polynomial Expressions of Carries in p-ary Arithmetics
This work addresses a specific mathematical problem in finite field arithmetic, with potential applications in cryptographic computation on encrypted data, and is incremental as it generalizes known results from binary to p-ary cases.
The paper tackles the problem of finding concrete polynomial expressions for carries in p-ary addition and multiplication, resulting in a new family of symmetric polynomials for addition and a simple formula with significantly fewer monomials for multiplication, specifically (n+1)(p-1)/2 + 1 monomials compared to the worst-case p^n.
It is known that any $n$-variable function on a finite prime field of characteristic $p$ can be expressed as a polynomial over the same field with at most $p^n$ monomials. However, it is not obvious to determine the polynomial for a given concrete function. In this paper, we study the concrete polynomial expressions of the carries in addition and multiplication of $p$-ary integers. For the case of addition, our result gives a new family of symmetric polynomials, which generalizes the known result for the binary case $p = 2$ where the carries are given by elementary symmetric polynomials. On the other hand, for the case of multiplication of $n$ single-digit integers, we give a simple formula of the polynomial expression for the carry to the next digit using the Bernoulli numbers, and show that it has only $(n+1)(p-1)/2 + 1$ monomials, which is significantly fewer than the worst-case number $p^n$ of monomials for general functions. We also discuss applications of our results to cryptographic computation on encrypted data.